Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C25 |
| direct product | C2×C11×#2061 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C11 |
| prime radical | C24×C11 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ C23×C4×C11 ◁ G | |
1130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37);(3,4,7,9)(5,10,6,8)(11,12,14,18)(13,20,21,16)(15,17,23,19)(22,26,24,25);(3,5,7,6)(4,8,9,10)(11,13,14,21)(12,16,18,20)(15,24,23,22)(17,26,19,25);(3,6,7,5)(4,8,9,10)(12,17)(15,23)(16,25)(18,19)(20,26)(22,24)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 16547);